THE SILVERMAN-TOEPLITZ THEOREM by BRIAN RUDER B, A,, Port Hays Kansas State College, 1961|. A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OP SCIENCE Department of Mathematics KANSAS STATE UNIVERSITY Manhattan, Kansas 1966 Approved by: Major Professor TABLE OP CONTENTS Page INTRODUCTION 1 CESARO SUMMABILITY 6 SILVERMAN-TOEPLITZ THEOREM 21 APPLICATIONS ' 22 ACKNOWLEDGEMENT 29 REFERENCES 30 INTRODUCTION The modern theory of sumraable series has its origin in the early eighteenth century. In 1713, a letter written by Leibniz to Christian Wolf discussing the series (1) 1 -1 +1-1+ ..., was published in the Acta Erudltorura. Leibnitz tried to Justify the value i for the series. He felt that the value i for the series was reasonable on the basis of his "law of continuity" and on the basis of the relation: (2) JL « 1 -x+x^x+ ..., 1+x which is true for values of x numerically less than unity. How- ever, Leibnitz felt that the value ^ should be obtainable from the series (1) without recourse to the series (2). One of his arguments for the value i was the following: If we take the sum of an even number of terms of the series (1), the value is al- ways zero; if we take the sum of an odd number of terms, the value is always unity. When we pass to the case of an infinite number of terms, there Is no reason to consider that we have either an odd or an even number of terms, and therefore no rea- son for assigning either the value unity or zero to the series, but rather it is reasonable to assign some Intermediate value. Moreover, in the process of allowing the number of terms to be- come Infinite, the values zero and unity for the series occur with equal frequency. He therefore considered it Justifiable, on the basis of probability, to assign to the infinite series a value intermediate between unity and zero, which is precisely their arithmetic mean, namely -|-. After this publication of Leibnitz, Euler started work in this area, and is credited with formalizing the theory of diver- gent series and suramability methods. Since that time, many prominent mathematicians have taken an interest in the subject and have developed the theory to what it is at the present time. Prior to a discussion of divergent series, suramability methods, and the Silverman-Toeplitz Theorem, it is essential that a few definitions be introduced as a means of communica- tions. In passing, one notes that the theory of convergent series is contained in the theory of divergent series in that convergence is nothing more than a logically simple method of assigning values to infinite series. The fact that there is more literature written on convergence than divergence is quite logical since there have been more elementary applications of convergent series. However this does not in any way detract from the interest and usefulness of the theory of divergent series. Definition 1 _ z An infinite series Z- a^ is a symbol for a definite n=0 sequence of complex numbers deducible from it, namely the se- quence of partial sums. The series 2^ 2^ n»0 " --i-^i^ ^^j^--- has n th partial sums (denoted by 3^) given by Definition 2 CO An infinite series 2L a^^ with an n th partial sura S^^, n=0 is said to be convergent, if lim S^ « S, and we call S the n-»«> "sum" of the series. Definition 3 aoL An infinite series 2^ a^ with n th partial sum S^^, is said to diverge definitely to + oo if lim S„ ■ +oo , and is said to diverge definitely to - oo if lim S_ » - oo n-*oo *i Definition h An infinite series ^ a„ with n th partial sura S„ is said to diverge indefinitely if it is neither convergent nor definitely divergent. The series (1), 1 -1 +1 - ..., is an example of an indefinitely divergent series. In the study of summability, the only divergent series of any interest, and the only ones that will be considered in this paper will be those which are indefinitely divergent. Definition ^ A summability method is a process which attempts to assign a number, complex or real, to infinite series. At this point, a few properties which are desirable in a summability method will be discussed. The first property which the method should possess is non-contradiction to the idea of convergence. (One wants the method to have this property, be- cause convergent series are of such great value, any summability method which contradicts it, could scarcely be expected to have much value.) Therefore one will want the method to satisfy a permanence condition. Definition 6 If an infinite series ^ a., converges to S, and if V- A2 o.-o ( 2^ ajj) «= S-j^, where V is a summability method, we say that the summability method satisfies the permanence condition. The sec- ond property that the method should possess is closely related to the permanence property and is called consistence. Definition 7 If for every infinite series ^ a„ which converges to S, we have V ( 2- ^^ ■ S, where V is a summability method, we say that the summability method V satisfies the consistency property. If a summability method satisfies both the permanence property and the consistency property, the following definition is appli- cable. Definition 8 A summability method is said to be regular if it satisfies the permanency property and it also satisfies the consistency property. There are other requirements that one might want a summability method to possess, such as an extension property, (which means that the summability method sums at least one diver- gent series) or a compatibility condition. (This property guar- antees that if an infinite series ^ aj^ is summable by two different methods, that both methods yield the same value.) Most methods do satisfy these properties, but in this paper only the property of regularity will be discussed, so no other prop- erties will be dealt with in detail. As should be quite evident from the definition of a regular summability method, any summability method that is not regular is of very little practical value. Since regularity is an impor- tant property, one would like to be able to determine whether an arbitrary summability method is regular or not. With the aid of the Silverman-Toeplitz Theorem, one is able to give a partial solution to this problem. If one is able to put a given summa- bility method into a form in which the Silverman-Toeplitz Theo- rem can be applied, then it is possible to determine whether the method is regular or not. Many summability methods can be put into this form, so the value of this theorem is evident. In this paper, the Silverman-Toeplitz Theorem will be proved and some of the applications of the Theorem will be given. Before discussing the Theorem, two definitions and examples of well known summability methods will be given which shall be referred to throughout the rest of this paper. CESARO SUMMA3ILITY Given the infinite series 2^ Uj^, Let Sj^ ■ ^ Uj^. n«0 k»0 Set C^ - Sq + S^ + S^ + ... + S^. Then one says that the given series is summable (C,l) to the sura C if 11m IL « c. n -♦<» n-fl This method of summability is due to Ernesto Cesaro. A more general Cesaro summability method is (C,K) summability, defined in the following manner: Sj^^°^ • S^ , and for k^l, s^(k) .so^^-i) .si(k.i) ^ ... .S^(k-l). (n » 1,2,3,...) and one now considers for each fixed k. If for some value of k, C^^^^ — » c, one says the sequence [sA is summable (C,K) to the sum C. In this paper, only (C,l) summability will be considered as it lends itself readily to examples. As an example of the application of the Cesaro method, note (k) . 3n"'' ^n (Tj that, with this method of summabllity, the series (1) is surnraable to the value g, which is consistent with Leibnitz's result. As a second example consider the problem of finding the analytic continuation of the infinite series ^ z^ which con- k-1 1-z^^l . — and verges to — for /zj ^1. For this series, S^ » — ~ C.<^+<^4.q4. «;- (1-z)^ + (1-2)^ -h ... + (1-z)" 1-z g ^ (l-z)(n+l 4- nz -f- (n-l)z + ... + (n-(n-l))z n "■ * 1-z Cn In considering the limit of , one finds that the limit n+1 exists if and only if jzjil; z / 1. To verify the case where z « -1, consider the partial sums, Sq - 1, S^ - 0, S^ - 1, ... , ^n - Sq + S^ + S^ + ... + S^, where S^ - ^ (1 + (-l)'^). Therefore, -^IL- . (n^D ^ hU ^ (-1)") . ^ ^ 1 ^ (-D" ^^^ n+1 2(n+l) l+(n+l) Cn 1 lim —. » ^. It should be noted that this value of i is in n ■"•** n+1 ■^ accordance with what one would expect to obtain if -1 were sub- stituted for X in the term , to which the original series 1-x converged. 8 Before Introducing the second summability method, the rela- tion between infinite series and their corresponding sequences will be discussed in a little more detail. As was mentioned earlier in this paper, to every series there corresponds a sequence of partial sums. It is also true that to every infinite sequence there corresponds an infinite series. The limit of the sequence is the same as the sum of the infinite series, when either value exists. Thus, in talking about summability methods, one is Justified in operating with the sequence of partial sums. The relation of a series to its sequence was introduced above, so at this point, only the relation of a sequence to its series will be mentioned. Given a sequence of complex numbers, (z^, z, , .,,) one can write this as a series by letting a^, « Zq, a^ - z^ - z., a2 - Zp - z, , ... ft " z^ - z^ , .... . One then has the series J- n n n- 1 / oe oo n-'O k«l The reason for introducing the relation of series to sequence is because the Silverraan-Toeplitz Theorem will be proved in the form where one considers a summability method as an operator on se- quences rather than an operator on infinite series. In this case one can consider a summability method as a transformation, which takes the given set of partial sums into a new set of partial sums. At this point the second method of summability, due to Euler, may be introduced. An infinite series Uq + u^^ + ... and its sequence Sq, Sj^, ,., of partial sums, are said to be summable to t by the Euler transformation (or method of summability) £(r) of order r, r being a complex constant, if t^ — ^ t as n — > oo where 'n - t„(r) - X (;:) '•' (i-)"-" s. • •ies ^ z^ . Consider this method when applied to the series ^^ z^ . From k«0 the definition of the transformation, one sees that the new se- quence, in this case, is given by: where k=0 ' 1-2 -^ — «s^- s z^ l-z ^ ^ n«0 One can write this transformation as ^ i(n|rMx-r,n-- . J (;;) (r.)Nl-., n-k k-0 k.Q n n Now note that k 2^ (M r^(l-r)^-^ . (r + 1-r)'^, and that J"/")- k«0 (rz)^ (l-r)"^-^ . (rz + (1-r))", and one can write the 10 transformation as ■'■ (1-z (rz + l-r)'^) . 1-z If z+l, then the series is summable to its analytic continuation if and only if | 1-r + rz | ^ 1; this is the same as C(r) .U^lzllK ^ . / r I fr/ If r«0 and r is fixed, the set of values of z for which the series is summable E(r) consists of the interior of the circle C(r) with center at the point and radius -^ . As a r (r( particular case consider r« — ^, then one has C(~L) « |z + ll;|^l5 for which the sequence is summable and this includes, and is larger than the original circle of convergence. From these two examples, one can note that the Euler method is more powerful than (C,l) summability in the case of the geo- metric series. However, (C,l) summability is more powerful than the Euler method when one deals with certain other series. Up to this point, nothing has been said about the regularity of these two methods. It will be shown (by making use of the Silverman-Toeplitz Theorem) that (C,l) summability is regular and that E(r) is regular if and only if certain restrictions are placed on r. The following Infinite matrices will be used as reference matrices throughout the rest of this paper. 11 u • s ^00 ^01 ^02 ^10 *11 ^12 ^20 ^21 ^22 ^nO ^nl ^n2 n n T U is an infinite matrix denoted by (a^^). S and T are infinite column matrices. The product U • S « T expresses a method of transforming the sequence [sj to the sequence (t |. In the problems to be discussed, the S matrix will have as elements the partial sums of the infinite series under consideration. The method of proof selected for this paper requires that a few lemmas be proven prior to consideration of the Silverman- Toeplitz Theorem. Lemma 1 eo 1 If the series ^ p^^q^ is convergent for every bounded n-0 es or merely for every convergent sequence Jpn{ , then the seri y -C_ qn is absolutely convergent. It is convenient, at this n«0 place to introduce the function signum, abbreviated sgn, which is defined as follows: 12 /i|i if z it sgn z ■ 1 if z « . Proof. If the sequence [vn] ^^ required to be bounded, then let Pjj » sgnq^ (which is obviously bovmded) and one has n=0 n=0 n=0 which is absolutely convergent and the first part of the lemma is proved. If the sequence jPn? i^ convergent, the following argument can be given. Assume ^ /QhI ^^ divergent. Let n»0 n ^n " Z-. /qyl , then as n — > a? approaches Infinity v=0 " monotonically. The series ^~ -^ is divergent, for by n«o ^n writing n+s n+s v=n v«n one notes that n+3 v»n ^ Ti+s Si+s 13 for s sufficiently large. Now set p « t—J and it is true ^n*^n Pj^ -» as n gets large since Q^ approaches infinity. Now by is ^ hypothesis ,^ Pn^^n converges, but n«0 y p „ . f JM . , . 2 I2nl which leads to a contradiction and therefore our assumption must 00 have been wrong and Z-Knl ^^ convergent and the lemma is proved. Lemma 1 shows that any condition on the U matrix given above which insures the existence of the T matrix for all bounded or convergent sequences [Sj^j, will have to include the condition: the sum of the absolute values of the elements of each row of U is finite, that is "*a nv v«0 exists for each n. As a matter of convenience, let V « / a » n ^ ' v»0 and W^ - ^ I &^^l for the rest of this paper. v«0 Lemma 2 A necessary and sufficient condition that the sequence /"t ? be bounded whenever the sequence fsl is bounded is that the sequence jw^? be bounded. 11^ Proof. Consider the sufficiency first (that is if fSy^f is bounded and jWj^f is bounded, then JT^f is bounded.) Let M be the least upper bound (abbreviated l.u.b.) of [Sj^), and let N be the l.u.b, of Wj^, then JT^^j ^ M-N and [t^J is bounded. For the necessity of the condition, note first of all that the elements of each column of U must be bounded. To show that this is true consider the set of sequences S « ^nv> ^ * 0»1»2, ... where ^^^ is the Kronecker delta defined as follows: fl if n « V "^ (o oti^erwise. Then one has T^ « a^^, n « 0,1,2, ••• and this sequence must be bounded for each v according to the hypothesis of the lemma. Now if Wj^ is unbounded, a bounded sequence yS^ f will be constructed for which {Tj^T is unbounded. Without loss of gen- erality, it may be assumed that W^^ tends monotonically to in- finity, for is this were not true, a subsequence ^Xj^T of W^^ for which this were true could be selected. Now define the sets of integers (u^), (k^^), (y^^) as follows: Uq - 0; k^ = 0; Jq is the smallest integer greater than k^ + 1 such that ^ /a /^ 1. v«yo The reason that one can do this follows from lemma 1, that is if the T matrix is to exist (and this is the only case to be con- sidered here then ^ l ^r,^rl must exist for all n and in v«0 15 particular for n » 0. If Z^ / ^nv / ©^Ists, then for v large enough, it is true that 2__ I *ov / ^ ■'■• ^°^ suppose that the first m of each of these sets, (u^), (k^), and (y^^) have been chosen; choose u^, k^, y^, as follows. First choose u^ as the smallest integer greater than u . for which there are numbers m-i k, greater than J^j^^i such that k ^^^ 2 /^uv/ ^ ~W^ - 1 for u - u^ . Now Uj^ exists since W^^ tends monotonically to infinity and the elements of each column of U are bounded, ie, la I ^ M(v) ' I nv ' ' ' f n « 0,1,2,..., V = 0,1,2,.,. , This means that once u__t has been chosen, one can select v large enough so that a Uj^ exists which satisfies (a) because the columns of U are bounded. Let ^m ■ ^m-l "•■ ^' ^m ^^ ^^®" ^ ^ ^°^ which (a) is true. Take y^^ as the smallest integer greater than k^^ + 1 such that CO for u « u_ , m v=y m This is possible by lemma 1 as was shown above. The sets (u^), (k^), and (y^) are completely defined and one has ^O^yQ^^l'^yi-^k^^y^-^k^-ry^ ^ ... . Now define the bounded sequence 7 S^ j for which JT ? will be 16 iinbounded, as otherwise ^yn-l-^~^n^* The sequence j S f is obviously bounded and one can see for -i V"0 ' / v=«0 *Siv ' v«0 Now ^0_ yj^_^ . ... J. v«0 v«k. ^ yn-1 n+1 ^n «^ Z v«0 ^"yn ^n ^^ wher-e | "u 1 ^ Z / ^uv / * 2- j\J- In defining j^^) many zeros were introduced and this is the reason for the strict inequality. Now to proceed, one has ^n yn-1 ^ yn-1 kn 02- ^"k^+1 v»0 v»y n Note that R is positive so that R > - R. Now replace R by - R in (1) and then use (2) to get 17 CO ^n-l v«=0 v=k+l k oo and W^ tends monotonically to infinity. The last inequality was obtained by using (a) and (b) , that is 00 -2 Z l^v|--2-l--2. v=yn Since the sequence l^rx) ^^ bounded, the sequence Iw ? is un- bounded, and the sequence Jt^J is unbounded, one can see that the condition that | W^ V be bounded is necessary, and lemma 2 is proved. Lemma 3 A necessary and sufficient condition that the sequence fT^? has a limit whenever the sequence |*S^? has the limit zero is that - ^ (a) 7^n) ^® bounded, i,e. W^ ^ W and. 16 (b) lim a^^^ = ly exists for each v. n —♦017 .tof T^» Z_ 1, Then the limit of '^xi' Z— \ \ ' Proof. Consider the sufficiency first. ex>_ k o» '^n ' 2i, ^nv \ ' Z. ^nv ^v + 2- ^nv ^y and v=0 v-0 v=k+l k oo K - Z ^nv Sv ^ Z l^uvl (S^l - lie W, ^^«^« J v=0 / v»k+l Lj^ ■ l.u.b. jS^j . Note that since the sequence Is (has limit 0, Ljj tends monotonically to 0, Now Fn- Z ^nv \j ^ Lj^W , so that k - ^ ^ - "^n - Z ^nv ^ ^ ^ W » ^° t^a* k k nv V k n Z nv v ic Let n approach infinity, and one has k k Z ly S^ - Lj^ W :^ lim_T^ - ill T^ ^ Z 1^ S^ + \ W. v-0 v-0 19 The sequence j T^V being bounded by lemma 2, these inequalities for k approaching infinity yield v»0 v=0 1 S :^ lim T^ ^ lim T^ ^ ^ 1 S„ V V n n Z— V v since L^^ — > 0, This proves the existence of lim T_, the con- n " Zi vergence of the series Z- 1^ S and the equation lim T, •o VM V V " wvi-«w^w*^ a.^m *j^ v=0 To show the necessity of the conditions, a method similar to the one used in lemma 2 will be employed. To show that the existence of lim a „ is a necessary condition, consider the set n * of sequences S^ « S^^ (Kronecker delta) n = 0,1,2,... . Then one has T^ - a^^^ and lim T^ exists by hypothesis. To show that the numbers W^^ must be bounded, consider the set of integers defined in lemma 2, but now choose with the given kj^, yj^, u ^ " 0,1,2,... ^sqn auv \ K ^n ^ V "^ ynJ « " "n otherwise. S^ is defined in this manner because the sequence Ss ? now has limit (since the sequence { W^ ^ is unbounded) and this is part of the hypothesis. The procedure is now exactly the same as in lemma 2 and for u « u_ l-ul^iRT. 20 which becomes infinite as n — >oo , and the proof is complete. Lemma Ij. A necessary and sufficient condition that the sequence I'^^i has the limit zero whenever the sequence /Sj^( has the limit zero is that (a) the sequence l^^] is bounded and, (b) lim a «0 v"012 '^••» Proof. The proof follows almost directly from lemma 3, From lemma 3 it is seen that CO li'" "^n ' Z. W » ^^«^« li'n ^nv ' \ * v*0 if lim Q.^^ - one has lim ^^ « 0. If lira ^^ = 0, then one can show lim a^^^ « in the following manner. Consider the sequence S^ « (0,0, 0,0, ...0,1,0,...) where the 1 is in the i th position. Then lim S. « 0. This says T^ « a . u - u but •^ i ui * n lira T^ - « 1^. The rest of the proof follows from lemma 3. Theorem 1 A necessary and sufficient condition that the sequence Un) ^^ ^ limit whenever the sequence Js^^j has a lirait is that (a) I W^^ is bounded, (b) n iilU a^^. 1^ V - 0,1,2,..., 21 oo ir,) 11m \/ „ lim / Ej^y exists v«0 Proof. If one applies lemma 3 to the sequence ? S " S f where S » 11m S^ then one has oo Za (S - S) « ^^^ (T - S V ) nv^ V ' n ^ n n' » v-0 CO so that IJ^ra T^ - s lira V^^ + ^L V^v ' ^^ v«0 and by condition (c) 11m V^ exists, so that If the sequence /s ? has a limit, T^^ also has a limit which Is given by 11m T^ - S 11m V^. SILVERMAN-TOEPLITZ THEOREM Theorem 2 A necessary and sufficient condition that the sequence )T T has a limit whenever the sequence 5Sjj? has a limit and that the two limits are equal Is that (a) 1^7 is bounded, <^) \ " ni^So ^nv -0, V - 0,1,2,... ^<5> nli« V„ - 1. n — ioo n 22 Proof. Prom Theorem 1, one can see that condition (a) is a neces- sary and sufficient to insure that )t I has a limit whenever ^Sj^ V has a limit. Also from Theorem 1, recall that lim Tjj - S lim Vj^ + 2_ 1^ ^\ " S) . v«0 If Iv -n'^'i ^nv " °' ^"^ ^^ I^'" ^n " ^ ^^^"^ ^^"^ "^n " ^ If lim Tj^ « S then lim T^^ •= S - S lim V^^ which says lim V^ - 1 The necessity of condition (b) follows from lemma \\.^ and the proof is complete, APPLICATIONS As the first application of this theorem, If will be shown that the (C,l) summability method is a regular method. Recall that the summability method was defined by lim _JL- where C^ - S^ + S, + ... S„. n — *oo n+1 n 1 n* Therefore, one has as the infinite matrix of transformation and the transformed set of sequences the following: 23 10 ^i-000 .. 2^ ^ L 3 3 3 _1 1 1_ n+1 n+1 n+1 n % So + Si 2 So + Si + sz 3 So + s, + ... ^n n+1 • • • Now consider the first condition of the theorem: j^Yii bounded. In this case Is CO VcQ and condition (a) is obviously satisfied. Condition (b) is: 1^ " ^^ Q-xiy - 0; V » 0,1,2,'..; for an arbitrary v one has ^nv ■ ;rrr ^^^ ^^^ -^ " O- condition (c) is: 11m V„ « 1. nv n+1 n-^oo n+1 n->oo n In this case one has lim V. CO n " lira JL. v«0 nv lim I i v-1 lim n (^) . 1. 2k From the fact that all three conditions of the Silverman-Toeplitz Theorem are satisfied, one knows that (C,l) summability is a regular method. As a second example, consider the Euler method of summa- bility. Recall that the method is defined for a fixed complex number r. Using the Silverman-Toeplitz Theorem, the values of r for which this method is regular shall now be determined. The Euler summability method is given by: n ^n " X ^nk Sk , k-0 where so that CO Or + (l-r))'' - "^ a^ k«l ^nk (2 J r^ (l-r)"-k, The infinite matrix of transformation is given by: 1 1-r r (l-r)*^ 2r(l-r) /, »n n /_ \n-l (l-r) ^ r(l-r) . . . n 3, «n-s ^ g r (l-r) To determine the set of r's for which the Euler method is regular 25 assume that the first condition of the Silverman-Toeplitz Theorem is satisfied, that is ) W^^ ( is bounded. From this it is seen CO *- ^ that Z— I a^^^ / is boxinded. For the Euler transformation, one v=0 m sees that 2_. j ^nv\ " U^l + |l-r( )"^, n - 0,1,2,... . v=0 In the case where r is an arbitrary complex constant, let r = a + bi and consider r «= Re^* where R = Va^ + b^ and © = arc tan — . Then a (/r| + |l-r|) = (R + 1 -R (cose + isine ) ) - (R +^1 - 2Rcose + R^) (R + Vl - 2R + R^ + 2R - 2Rcos© ) R . (1-R) \/l^ 2^il^so2 , » (1-R)2 Since 1 - cos© 2 0, \l + ^^(^'^ose ) ^ ^^ ^^^ y (1-R)2 (|r| + (l-r|) > R + (1-R) .1-1, and equals 1 if and only if © = 2k7r ; k « 0,1,2,... . If ©^ 2krr , then { \r] + /l-r/) > 1, and one can readily see that condition (a) is not satisfied since (Irj + |l-r/) -» 00 as n -^oo . Therefore, if condition (a) of t he theorem is to be satisfied 26 one must have 9 = 2k 7J, and this says that r must be real. If r is real then condition (a) is satisfied if and only if ^ r £ 1. After consideration of the first condition, r has been re- stricted to be real and in the interval ^ r 6 1, if the Euler method of summability is to be regular. The two cases where r = and r = 1 will be considered separately. For these values the infinite matrices of transformation are given by: 10 0... 10 0. . . 10 0.. • • • • • « • 10 10 10 0' • '010 n 'n 'n 27 The first set of matrices are those obtained for r = 0, and the second for the case where r « 1. In the first case, condition (b) of the Silverman-Toeplitz Theorem is not satisfied for v » 0, and therefore when r >= 0, the Euler method is not regular. In the second case where r = 1, one can see that this is Just the identity transformation, which is obviously regular. For the last two conditions of the theorem, one needs only to consider r real and in the interval ^ r < 1. Consider con- dition (b) ni^^ a^v •= 0, for v = 0,1,2,... . To show that this condition is satisfied, consider the infinite series whose nth term is the a^^^ of condition (b) . It will be shown that the series converges and therefore that the nth term approaches 0, which in turn will show that ^^^^ a^^ - 0. Consider the in- finite series I n+v , „ ,_ .n-v rv (i-r)' where v is fixed. Now apply the ratio test to this, and one has lim n -*» f n+l+k) . ,n+l-k w (l-r) n-k lim n -><» 1 + n^ H = )-/• For <«i r ^ 1, this series converges and therefore the nth term approaches and ^^^ a =0. n — $» nv • Condition (c) is obviously satisfied since it has been shown 28 that n v=0 but since r>0 this is { r + l-r)" «= (1)" = 1. The conclusions that one can draw from the Silverman-Toeplitz Theorem concerning the Euler method of summability is that the method is regular if and only if r is real and in the interval < r ^ 1. Prom these two examples, the importance of the Silverman- Toeplitz Theorem as a tool in the theory of divergent series is evident. It can be used to determine whether a given summability method is regular. In certain cases, It can be used to determine the values for which a method is regular. In either case, the Theorem gives much information and also leads to many interesting mathematical problems. ACKNOWLEDGEMENT The writer wishes to express his sincere appreciation to Dr. C, P. Koch for introducing him to this subject and for his patient guidance and supervision given during the preparation of this report. REFERENCES Kaplan, Wilfred. Advanced Calculus. Addison-Wesley Publishing Co., 19^2, PP. 350-375. Knopp, Konrad. Theory and Applications of Infinite Series. New York: Hafner Publishing Co., 1928, pp. kSl-k^S. Moore, Charles N. Summable Series and Convergent Factors. New York: American Mathematical Society Colloquium Publications, Volume 22, 1938, pp. 1-10. Szasz, Otto. Introduction to the Theory of Divergent Series. Stechert- Hafner, Inc., 1914^, pp. 1-29. THE SILVERMAN-TOEPLITZ THEOREM by BRIAN RUDER B, A., Port Hays Kansas State College, I96I4. AN ABSTRACT OP A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MSTER OP SCIENCE Department of Mathematics KANSAS STATE UNIVERSITY Manhattan, Kansas 1966 The modern theory of summability inethods and divergent series had its start in the eighteenth century. Some of the first men to work in this area were Leibnitz and Euler. In this paper methods for assigning values to certain diver- gent series are discussed. If these methods are to be of any value, certain restrictions must be placed on them. One of the more Important conditions that the method must possess is called regularity. To say that a summability method is regular means that the method "sums" convergent series to the value that the series converges to in its circle of convergence. The Silverraan- Toeplitz Theorem supplies three necessary and sufficient condi- tions that a method must satisfy if it is to be regular. A proof of the Silverman-Toeplitz Theorem is given in this paper. After the Theorem is proved, two applications of the Theorem are given.